2.4.2 The Multiplication Rule for union of two sets

The definition of conditional probability yields the following result,

• obtained by multiplying both sides of Equation (2.3) in defintion of conditional probability by $P\left(B\right)$ .

The Multiplication Rule

$$P\left(A\cap B\right)=P\left(A\mid B\right)\cdot P\left(B\right)$$

• This rule is important because it is often the case that
  • $P\left(A\cap B\right)$ is desired,
  • whereas both $P\left(B\right)$ and $P\left(A\mid B\right)$ can be obtained from the problem description.
• Consideration of $P\left(B\mid A\right)$ gives $P\left(A\cap B\right)=P\left(B\mid A\right)\cdot P\left(A\right)$

EX 2.27 blood type

• The multiplication rule is most useful when
  • the experiment consists of several stages in succession.
• The conditioning event $B$ then describes the outcome of the first stage
  • and $A$ the outcome of the second,
  • so that $P\left(A\mid B\right)$ (conditioning on what occurs first) will often be known.
• The rule is easily extended to experiments involving more than two stages.

Example

Consider three events $A_1$, $A_2$, and $A_3$.

• The triple intersection of these events can be represented as the double intersection $\left(A_1\cap A_2\right)\cap A_3$ .
• Applying our previous multiplication rule to this intersection
• and then to $A_1\cap A_2$ gives

\begin{align} &P\left( {{A}{1} \cap {A}{2} \cap {A}{3}}\right) \ = &P\left( {{A}{3} \mid {A}{1} \cap {A}{2}}\right) \cdot P\left( {{A}{1} \cap {A}{2}}\right) \ = &P\left( {{A}{3} \mid {A}{1} \cap {A}{2}}\right) \cdot P\left( {{A}{2} \mid {A}{1}}\right) \cdot P\left( {A}{1}\right) \end{align}

$$-Thusthetripleintersectionprobabilityisaproductofthreeprobabilities,-twoofwhichareconditional.$$

EX 2.28 blood type again

Remark

• When the experiment of interest consists of a sequence of several stages,
  • it is convenient to represent these with a tree diagram.
• Once we have an appropriate tree diagram,
  • probabilities and conditional probabilities can be entered on the various branches;
• this will make repeated use of the multiplication rule quite straightforward.

EX 2.29